# Fence with minimum cost

A fence is to be built to enclose a rectangular area of 300 square feet. The fence along three sides is to be made of material that costs 3 dollars per foot, and the material for the fourth side costs 12 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.

## 1 Answer

Let $x$ and $y$ be the sides oif the rectangular region. Then

\[xy=300 \Rightarrow y=\frac{300}{x}.\]

The cost of the material is

\[C=3(x+y+x)+12 y=6x+15y=6x+15\frac{300}{x}.\]

So

\[C=6x+\frac{4500}{x}.\]

To minimize $C$ we take a derivative to find the critical points

\[C'(x)=6-\frac{4500}{x^2}=0\]

\[\Rightarrow x^2=\frac{4500}{6}=750.\]

Hence

\[x=\sqrt{750} \text{and} y=\frac{300}{\sqrt{750}},\]

are dimensions of the enclosure that is most economical to construct.

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